While the regression analyses highlighted updating and intelligence as the most important predictors, inhibition, switching, the visuospatial sketchpad, and
phonological loop were all independent predictors. As the structures of working memory and executive function are still debated, and it is difficult to measure these constructs, it would be cogent to consider a model which included all these
constructs (updating, inhibition, switching, visuospatial sketchpad and phonological loop) in one latent variable (see Figure 7.4). This model displayed a good fit,
although factor loadings were stronger for visual constructs. To address whether modality was impacting the association between cognitive constructs and arithmetic ability, this single latent factor was divided into two modality specific latent variables (see Figure 7.5), both of which displayed a good fit. In all models each construct made a significant contribution to the model.
This is an important finding as evidence for switching and inhibition as important predictors of mathematical ability is mixed, with significant relationships found in some studies (e.g., Blair & Razza, 2007; Cantin et al., 2016; Yeniad et al., 2013), but not others (Cragg et al., 2017; Lee et al., 2012; Van der Ven et al., 2012). Research which predominantly utilises regression analyses has suggested that inhibition and switching explain unique variance when studied independently, which is then accounted for by working memory (updating, phonological loop and visuospatial sketchpad) when it is added to the model (Bull & Lee, 2014; Cragg et al., 2017; Lee & Bull, 2015). Alternatively, inhibition and switching account may account for less
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variance in mathematical ability than working memory (Cragg et al., 2017; Friso-van den Bos et al., 2013).
In this study inhibition and switching in both modalities were found to be significant components of both a single executive functioning latent variable, and modality specific latent variables. Indeed, factor loadings for visual inhibition and switching were among the highest in both models, and when arithmetic ability was added to each model (see Figures 7.6 and 7.7), they were the highest. Conversely, loadings for auditory inhibition and switching were the weakest, (although still significant), a possible indication they make a lesser contribution to arithmetic achievement. However, before drawing this conclusion, in this study there is the possibility findings may have been impacted by tasks used to index these constructs having low construct validity.
Evidence for the importance of the visuospatial sketchpad and phonological loop as predictors of mathematical ability is inconsistent, with suggestions that links may be age specific, with the phonological loop having more impact early in development, and the visuospatial sketchpad becoming increasingly important as mathematics becomes more complex (e.g., Meyer et al., 2010). However, there is also evidence for the phonological loop being an important predictor in middle childhood
(Andersson, 2008).
This study confirmed both the visuospatial sketchpad and phonological loop as independent predictors, which remained when age was accounted for. However, examining them alongside other predictors resulted in both losing predictive power, which was also true when modality specific predictors were investigated. This result is in line with much research (e.g., Geary, 2011), but contrary to others including a meta-analysis (Friso-van den Bos et al., 2013).
The reasons for the loss of predictive power may be understood by looking in more detail at the structural equation modelling (see Figures 7.4 to 7.11), where the visuospatial sketchpad always covaried with visual updating and in more complex models the phonological loop covaried with auditory updating. Covariance between updating and its modality appropriate slave component is not surprising, nor, given the age range of this study, is the finding that the ability to hold information in memory and also manipulate it, explains additional variance.
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8.2.3 Possible models for representing executive function and working memory and intelligence.
This study investigated two ways of modelling executive function and working memory constructs. The first included all executive functioning and working memory predictors in a single latent variable (see Figure 7.4), whilst the second split them by modality, hence modelled visual executive function and working memory, and
auditory executive function and working memory latent variables (see Figure 7.5). Each model displayed a good fit, which was also the case when arithmetic was added to the model (see Figures 7.6 and 7.7). Importantly each latent variable was a strong predictor of arithmetic ability.
Placing the modality specific latent executive function variables into a model with verbal, and non-verbal intelligence facilitated an investigation into their association with arithmetic ability (see Figure 7.8). In the resultant model direct paths to
arithmetic ability were observed from each modality specific executive functioning and working memory variables solely. Paths from verbal and non-verbal intelligence were indirect, through both executive function and working memory latent variables. However, whilst there was a significant path between each modality specific latent variable there was a covariance between auditory updating and the visual latent variable, possibly suggesting that whilst each modality is impacting arithmetic ability, both are important as there are interactions between them.
Representing the executive function and working memory predictors as a single latent variable (Figure 7.9) and placing it in a model with the intelligence predictors, also resulted in a direct path from the latent variable solely, with paths from the intelligence predictors once again being indirect through the latent variable.
When age was included in the two latent variable model (see Figure 7.10), links to arithmetic ability for both intelligence predictors were once again through each executive functioning latent variable. Interestingly, for both verbal and non-verbal intelligence there was a stronger link to the auditory executive function and working memory latent variable. Adding age to the single latent variable model (see Figure 7.11) produced similar relationships, however, there was now a direct path between non-verbal intelligence and arithmetic ability.
In the two latent variable model the link between age and arithmetic ability was indirect through verbal intelligence (B = .87), non-verbal intelligence (B = .66), and visual executive functioning and working memory (B = .43). This finding was
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replicated in the single latent variable model. Age has been highlighted as a possible confound, with evidence for this phenomenon in general cognitive abilities, and numerical acuity literature (Fazio et al., 2014; Geary, 2011; Meyer et al., 2010), although results are inconsistent (e.g., Chen & Li, 2014). In this study findings appear to converge to age having an indirect influence on arithmetic ability, through its impact on other cognitive abilities. Specifically, age appears to impact measures of intelligence more than executive functioning, with the strength of the impact influenced by modality, as age affected verbal intelligence to a greater extent than non-verbal intelligence, and there was a direct path to the visual executive
functioning and working memory latent variable solely.
In the two latent variable model age impacting visual executive functioning but not auditory executive functioning is intriguing. It may be due to auditory processes maturing earlier than visual processes. However, given the paucity of tasks designed to measure auditory inhibition and switching, particularly across a wide age range, results may be due to these tasks displaying lower construct validity. As this study highlights the importance of auditory executive functioning it seems appropriate for reliable measures for auditory switching and inhibition appropriate across
development to be developed.
Each of the three models investigated (updating, single executive function and working memory latent variable and two modality specific latent variables) have highlighted the importance of executive functioning and modality in predicting
arithmetic ability, above and beyond the impact of other constructs. In deciding which is the preferred model, the updating one is not optimal, given the latent variable is comprised of just two variables. Relationships in the other two are very similar, with the single latent variable being more parsimonious. However, modelling as two modality specific latent variables is valuable, as it highlights the importance of
considering modality when investigating the construct that underpin arithmetic ability, as there are significant direct paths from both the visual and auditory latent variables.
The alternative analysis utilised executive function and working memory variables which had been adjusted to account for the effect of age and verbal and non-verbal intelligence. Finding were similar to the main analysis; executive function and working memory predictors in both modalities having a significant impact on arithmetic ability. However, placing all executive function and working memory predictors into a single latent variable with arithmetic produced a model which
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displayed a poor fit, and placing two modality specific latent variables into a single model with arithmetic did not converge. Modality specific latent variables consisting of combinations of three of the executive function and working memory predictors did produce models which displayed a good fit and significant path to arithmetic ability. It is possible that the poor fit or lack of convergence of the more complex model was due to sample size, however, the models that did display a good fit, also point to executive functioning constructs in both modalities having an impact on arithmetic ability above and beyond the effect of age and verbal and non-verbal intelligence.
8.2.4 Numerical acuity.
Congruent with much research, visual numerical acuity was found to be an independent predictor of arithmetic ability even accounting for age (Fazio et al., 2014; Halberda & Feigenson, 2008; Schneider et al., 2017). However, visual
numerical acuity was indexed by incongruent (where numerical and visual properties were incongruent) trials solely, as the link between arithmetic and congruent trials was insignificant.
Entering visual numerical acuity concurrently into a regression analysis with other predictors, resulted in it losing predictive power, hence congruent with previous findings, the link between arithmetic ability and visual numerical acuity appears to be at least partly driven by other cognitive abilities (Gilmore et al., 2013; Price et al., 2012).
Recently it has been suggested that much of the research investigating visual numerical acuity is underpowered (Chen & Li, 2014). This study addressed this by utilising a task which consisted of 168 trials, across two sessions. Reliability across the two sessions was found to be good (Cronbach’s a = .85).
The auditory numerical acuity task was also included in each session and displayed good reliability (Cronbach’s a = .87). It was an independent predictor of arithmetic ability, which remained when accounting for age, however, it became insignificant when entered into a regression analysis with other cognitive abilities. As very little research has utilised auditory numerical acuity, there is little to compare this result to, however regression coefficients were similar in magnitude for visual and auditory numerical acuity (see Table 8.1), which may provide additional evidence for this ability being multimodal (Arrighi et al., 2014; Izard et al., 2009).
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Although it should be noted that zero and age-corrected correlations between these constructs were less than .8 (see Tables 6.7, 6.8, 9.5 and 9.6).
Table 8.1
Regression Coefficients for Visual and Auditory Numerical Acuity
Regression Model Auditory numerical acuity Visual numerical acuity
Independent .59 .60
Age-corrected independent .21 .22
All predictors -.05 -.02
8.3 Summary
Phase one of this study looked to determine the cognitive abilities which are predictive of arithmetic ability in the general population and whether predictive power is impacted by the modality of stimuli presentation. While the way data were
analysed produced differing results, the overarching finding was that executive function and working memory constructs had a significant impact on arithmetic ability, even when other predictors; age, verbal intelligence and non-verbal intelligence, were accounted for. A second key finding was the need for future research to overtly consider the modality in which stimuli are presented, as both auditory and visual predictors had an impact on arithmetic ability.
The following chapters will consider data and findings from phase two of this study.
Phase Two: Results and Discussion
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